SLIDE 1
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✷
❛①✐s ♦❢ t❤❡ ❛r♠✱ X✱ ✐♥ t❤❡ ❢♦r♠✱ ∂R
∂X ∂d ∂X
- =
- R
d 1 ˆ K Γ
- .
P - - PDF document
P - - PDF document
P strt ts t
SLIDE 2
SLIDE 3
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✸
✷✳ ❖❝t♦♣✉s✬s ❆r♠ P❤②s✐♦❧♦❣②✿ ❆♥ ❖✈❡r✈✐❡✇ ❖r❣❛♥s s✉❝❤ ❛s t❤❡ ♠❛♠♠❛❧✐❛♥ t♦♥❣✉❡✱ t❤❡ ❡❧❡♣❤❛♥t✬s tr✉♥❦ ❛♥❞ t❤❡ ♦❝✲ t♦♣✉s✬s ❛r♠s ❛r❡ t❡r♠❡❞ ▼✉s❝✉❧❛r ❍②❞r♦st❛ts ❬✶✶❪✳ ❚❤❡② ❛r❡ ❝❤❛r❛❝t❡r✐③❡❞ ❜② t❤❡✐r ❧❛❝❦ ♦❢ ✈❡rt❡❜r❛s ❛♥❞ ❝♦♠♣r❡ss✐❜❧❡ ❝❛✈✐t✐❡s✳ ❚❤❡ ♠♦st ✐♠♣♦rt❛♥t ❢❡❛t✉r❡ ♦❢ ♠✉s❝✉❧❛r ❤②❞r♦st❛ts ✐s t❤❡✐r r❡❧❛t✐✈❡❧② ❧❛r❣❡ ❜✉❧❦ ♠♦❞✉❧✉s t❤❛t r❡✲ s✉❧ts ❢r♦♠ ❛ ❞❡♥s❡ ♠✉s❝✉❧❛t✉r❡ ✇✐t❤♦✉t ❛♥② ❣❛s✲✜❧❧❡❞ ❝❛✈✐t✐❡s ♦r ❧❛r❣❡ ❜❧♦♦❞ ✈❡ss❡❧s ❬✶✷❪✳ ❚❤✐s ❡♥❛❜❧❡s ♠❛♥✐♣✉❧❛t✐♦♥ ♦❢ ❛♥ ♦r❣❛♥ ❧❛❝❦✐♥❣ ❛♥② ✈❡rt❡❜r❛t❡ s❦❡❧❡t♦♥ ❜② ❛❝t✐✈❛t✐♥❣ t✇♦ ♦r ♠♦r❡ ♠✉s❝❧❡ ❣r♦✉♣ s✐♠✉❧t❛♥❡♦✉s❧②✳ ❚❤❡ ♦❝t♦♣✉s✬s ❛r♠ ❝♦♥s✐sts ♦❢ t❤r❡❡ ♣r✐♠❛r② ♠✉s❝❧❡ ✜❜❡r ❣r♦✉♣s s✉rr♦✉♥❞✲ ✐♥❣ ❛ ❝❡♥tr❛❧ ❛①✐❛❧ ♥❡r✈❡ ❝♦r❞ ✭❋✐❣✉r❡ ✷✳✶✮✿ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♠✉s❝❧❡s✱ t❤❡ tr❛♥s✈❡rs❡ ♠✉s❝❧❡s ❛♥❞ t❤❡ ♦❜❧✐q✉❡ ♦r ❤❡❧✐❝♦✐❞❛❧ ♠✉s❝❧❡s✳ ❚❤❡ ❧❛tt❡r ❛♣♣❡❛r ✐♥ ❜♦t❤ ❛ r✐❣❤t ❤❛♥❞❡❞ ❝♦✐❧ ❛♥❞ ❛ ❧❡❢t ❤❛♥❞❡❞ ❝♦✐❧✳ ❚❤❡ tr❛♥s✈❡rs❡ ♠✉s❝❧❡ ✜❜❡rs ❛r❡ ♦r✐❡♥t❡❞ ✐♥ ♣❧❛♥❡s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ t❤❡ ❛①✐s ♦❢ t❤❡ ❛r♠✳ ❚❤❡② ❛r❡ ❧❛✐❞ ✐♥ ❛♥ ♦rt❤♦❣♦♥❛❧ ❛rr❛② s✉rr♦✉♥❞✐♥❣ t❤❡ ❛①✲ ✐❛❧ ♥❡r✈❡ ❝♦r❞✳ ❚✇♦ ❜✉♥❞❧❡s ❡①t❡♥❞ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❧❛t❡r❛❧ ♣❧❛♥❡✱✶ ❛♥❞ t✇♦ ❜✉♥❞❧❡s ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ❢r♦♥t❛❧ ♣❧❛♥❡ ✭s❡❡ ❋✐❣✉r❡ ✸✳✶✮✳ ❚❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ✜❜❡rs s✉rr♦✉♥❞ t❤❡ tr❛♥s✈❡rs❡ ✜❜❡rs ✐♥ ❢♦✉r ❜✉♥❞❧❡s✱ ❛♥ ♦r❛❧ ❜✉♥❞❧❡✱ ❛♥ ❛❜♦r❛❧ ❜✉♥❞❧❡ ❛♥❞ t✇♦ ❧❛t❡r❛❧ ❜✉♥❞❧❡s✳ ❚❤❡ ❝r♦ss s❡❝t✐♦♥ ❛r❡❛ ✐s ❧❛r❣❡r ✐♥ t❤❡ ❛❜♦r❛❧ ❜✉♥❞❧❡✱ ✐♥ ❝♦♠♣❛r✐s♦♥ ✇✐t❤ t❤❡ ♦r❛❧ ❛♥❞ ❧❛t❡r❛❧ ❜✉♥❞❧❡s✳ ❚❤✐s ❡♥❛❜❧❡s t❤❡ ❡①❡rt✐♦♥ ♦❢ ❤✐❣❤❡r ♠♦♠❡♥ts ✇❤❡♥ t❤❡ ❛r♠ ✐s ❜❡♥t ❛❜♦r❛❧❧② t♦ r❡✈❡❛❧ t❤❡ s✉❝t✐♦♥ ❧✐♥❡✳ ❍❡❧✐❝♦✐❞❛❧ ♠✉s❝❧❡ ✜❜❡rs ❛♣♣❡❛r ✐♥ t❤r❡❡ ❞✐✛❡r❡♥t ❧❛②❡rs✿ ✐♥t❡r♥❛❧✱ ♠❡❞✐❛♥ ❛♥❞ ❡①t❡r♥❛❧✳ ■♥ ❡✈❡r② ❝r♦ss s❡❝t✐♦♥✱ t❤❡ t❤r❡❡ ❧❛②❡rs ✭♦r ❣r♦✉♣s✮ s♣✐r❛❧ ❛r♦✉♥❞ t❤❡ ❝❡♥t❡r❧✐♥❡ ❜♦t❤ ✐♥ ❛ r✐❣❤t ❤❛♥❞❡❞ ❤❡❧✐① ❛♥❞ ❛ ❧❡❢t ❤❛♥❞❡❞ ❤❡❧✐①✳ ❑✐❡r ❛♥❞ ❙t❡❧❧❛ ❡①❛♠✐♥❡❞ ✐♥ ❬✶✷❪ t✇♦ ♦❝t♦♣✉s✬s s♣❡❝✐❡s ❛♥❞ r❡♣♦rt❡❞ ♠❡❛♥ ♣✐t❝❤ ❛♥❣❧❡ ♦❢ 62◦ ❢♦r ❡①t❡r♥❛❧ ❛♥❞ ♠❡❞✐❛♥ ♦❜❧✐q✉❡ ♠✉s❝❧❡s ✜❜❡rs✳ ■♥t❡r♥❛❧ ♦❜❧✐q✉❡ ♠✉s❝❧❡s ✜❜❡rs ❤❛✈❡ ❛ ❧♦✇❡r ♠❡❛♥ ♣✐t❝❤ ❛♥❣❧❡ t❤❛t ✈❛r✐❡s ❜❡t✇❡❡♥ t❤❡ t✇♦ s♣❡❝✐❡s✿ 42◦ ❢♦r ❖❝t♦♣✉s ❜r✐❛r❡✉s ❛♥❞ 56◦ ❢♦r ❖❝t♦♣✉s ❞✐❣✉❡t✐✳ ❆s t❤❡ ❛r♠ ❞♦❡s ♥♦t ❝♦♥t❛✐♥ ❛♥② r✐❣✐❞ s❦❡❧❡t♦♥✱ ❝♦♥tr♦❧ ♦❢ t❤❡ ❝♦♥✜❣✉✲ r❛t✐♦♥ ✐s ♠❛❞❡ ♣♦ss✐❜❧❡ ❜② ❝♦♠❜✐♥✐♥❣ ✐♥❝♦♠♣r❡ss✐❜✐❧✐t② ✇✐t❤ ❝♦♥tr❛❝t✐♦♥s ♦❢ ❛ ♥✉♠❜❡r ♦❢ ♠✉s❝❧❡ ❣r♦✉♣s✳ ❋♦r ❡①❛♠♣❧❡✱ ❛ ❝♦♥tr❛❝t✐♦♥ ♦❢ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ♠✉s❝❧❡ ❛t t❤❡ ♦r❛❧ s✐❞❡ ✇✐❧❧ ❝❛✉s❡ s❤♦rt❡♥✐♥❣ ♦❢ t❤❡ ❛r♠ ❛♥❞ ❛♥ ✐♥❝r❡❛s❡ ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥ ❛r❡❛✳ ❚♦ ❛✈♦✐❞ t❤❡ ❝♦♥tr❛❝t✐♦♥ ❛♥❞ ❝r❡❛t❡ ✢❡①✉r❡✱ t❤❡ ❝r♦ss s❡❝t✐♦♥ ❛r❡❛ ✐s ❤❡❧❞ ✜①❡❞ ❜② ❝♦♥tr❛❝t✐♥❣ t❤❡ tr❛♥s✈❡rs❡ ♠✉s❝❧❡s✳ ❆s ❛ r❡s✉❧t ♦❢ t❤❡ ❛r♠✬s ✐♥❛❜✐❧✐t② t♦ ❝❤❛♥❣❡ ✐ts ✈♦❧✉♠❡✱ t❤❡ ❛❜♦r❛❧ s✐❞❡ ♠✉st ❡❧♦♥❣❛t❡ ❛♥❞ t❤✉s ♣r♦❞✉❝❡ ✢❡①✉r❡ ♦❢ t❤❡ ❛r♠ ✐♥ t❤❡ s❛❣✐tt❛❧ ♣❧❛♥❡✳ ✸✳ ❈♦♥❢✐❣✉r❛t✐♦♥s ♦❢ t❤❡ ❆r♠ ✸✳✶✳ ◆♦t❛t✐♦♥ ❛♥❞ Pr❡❧✐♠✐♥❛r✐❡s✳ ❚❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ❛♥ ❡❧❧✐♣t✐❝❛❧ ❝②❧✐♥❞❡r ✐♥ t❤❡ ✈❡❝t♦r s♣❛❝❡ R3✳ ❊❛❝❤ ♠❛t❡r✐❛❧ ♣♦✐♥t ✐♥ t❤❡ ❛r♠ ✐s ❞❡s❝r✐❜❡❞ ❜② t❤❡ r❡❢❡r❡♥❝❡ ❝♦♦r❞✐♥❛t❡s (X1, X2, X3) =
✶◆♦t❡ t❤❛t ✐♥ ♦r❞❡r t♦ s❤♦✇ t❤❡ s✉❝t✐♦♥ ❡❧❡♠❡♥ts ❝❧❡❛r❧②✱ t❤❡ ❧❛t❡r❛❧ ♣❧❛♥❡ ✐s ❞r❛✇♥
✈❡rt✐❝❛❧❧② ✐♥ ❋✐❣✉r❡ ✸✳✶✳
SLIDE 4
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✹
Nerve cord Transverse fibers Oblique fibers
G2 G1 Suction elements line
Longitudinal fibers
Oral side Aboral side
Frontal Plain
❋✐❣✉r❡ ✷✳✶✿ ❆ s❝❤❡♠❛t✐❝ ❝r♦ss s❡❝t✐♦♥ ♦❢ ❛♥ ♦❝t♦♣✉s✬s ❛r♠✳ Y X S
Suctions line
S = 1
Sagittal Plain Frontal Plain
Arm’s axis
❋✐❣✉r❡ ✸✳✶✿ ❚❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠
(X, Y, S) ✐♥ s♦♠❡ r❡❢❡r❡♥❝❡ ❢r❛♠❡ ❛♥❞ ✐t ❛ss✉♠❡❞ t❤❛t ❛t t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✲ ✜❣✉r❛t✐♦♥✱ t❤❡ ❝❡♥t❡r❧✐♥❡ ♦❝❝✉♣✐❡s t❤❡ ♣♦✐♥ts (0, 0, S) ❢♦r S ∈ [0, 1] ✇✐t❤ t❤❡ ❜❛s❡ ♦❢ t❤❡ ❛r♠ ❜❡✐♥❣ ❧♦❝❛t❡❞ ❛t S = 0✳ ❚❤✉s✱ t❤❡ ❝❡♥t❡r❧✐♥❡ ♦❢ t❤❡ ❛r♠ ✐s s✐t✉❛t❡❞ ❛❧♦♥❣ t❤❡ X3 = S ❛①✐s ❛♥❞ ✐s s❡t t♦ ❜❡ ♦❢ ❛ ✉♥✐t ❧❡♥❣t❤ ❢♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✳ ❚❤❡ ♣r✐♥❝✐♣❛❧ ❛①❡s ♦❢ t❤❡ ❡❧❧✐♣t✐❝❛❧ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❝②❧✐♥❞❡r ❛r❡ ❞❡♥♦t❡❞ ❛s a0 ❛♥❞ b0 ❛♥❞ ❛r❡ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ X ❛♥❞ Y ❝♦♦r❞✐♥❛t❡ ❛①❡s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤❡ s✉❝t✐♦♥ ❡❧❡♠❡♥ts ❛r❡ ❧♦❝❛t❡❞ ♦♥ t❤❡ ♣♦✐♥ts ♦♥ t❤❡ ❝✐r❝✉♠❢❡r❡♥❝❡ ♦❢ t❤❡ ❝②❧✐♥❞❡r ❢♦r ✇❤✐❝❤ Y = 0 ❛♥❞ X = a0 ✭s❡❡ ❋✐❣✉r❡ ✸✳✶✮✳ ❚❤❡ r❛❞✐✉s ✈❡❝t♦r ✐♥ t❤❡ r❡❢❡r❡♥❝❡ ❢r❛♠❡ t♦ t②♣✐❝❛❧ ♠❛t❡r✐❛❧ ♣♦✐♥t ♦❢ t❤❡ ❛r♠ ✐s R = R (X, Y, S) ❛♥❞ t❤❡ ✉♥❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ❝✉r✈❡ ✇✐❧❧ ❜❡ ❞❡♥♦t❡❞ ❛s R0(S) = R (0, 0, S)✳ ❆t ❡❛❝❤ ♣♦✐♥t ✐♥ t❤❡ r❡❢❡r❡♥❝❡ st❛t❡ ✇❡ ♠❛② ❞❡✜♥❡ t❤❡ ❜❛s❡ ✈❡❝t♦rs✱ Gp = ∂R
∂Xp . ❆s t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❛ r✐❣❤t ❝②❧✐♥❞❡r✱
t❤❡ ✈❡❝t♦rs {Gp} ❛r❡ ♦rt❤♦♥♦r♠❛❧ ❛♥❞ ❛r❡ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ ✉♥✐t ✈❡❝t♦rs ❛❧♦♥❣ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♦r❞✐♥❛t❡ ❛①❡s✳ ❚❤❡ ❛❝t✉❛❧ ❝♦♥✜❣✉r❛t✐♦♥s ♦❢ t❤❡ ❛r♠ t❛❦❡ ♣❧❛❝❡ ✐♥ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡ ✇❤✐❝❤ ✇❡ ❞♦ ♥♦t ♥❡❝❡ss❛r✐❧② ✐❞❡♥t✐❢② ✇✐t❤ t❤❡ r❡❢❡r❡♥❝❡ ❢r❛♠❡✳ ❚❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡
SLIDE 5
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✺
✐s r❡♣r❡s❡♥t❡❞ ❜② ❛ ✸✲❞✐♠❡♥s✐♦♥❛❧ ❊✉❝❧✐❞❡❛♥ s♣❛❝❡ ❛♥❞ ✐t ✐s ❛ss✉♠❡❞ t❤❛t ❛ s♣❡❝✐✜❝ ♦rt❤♦♥♦r♠❛❧ ❢r❛♠❡ ✐s ❣✐✈❡♥✳ ❚❤✉s✱ ❞❡♥♦t✐♥❣ t❤❡ ♦rt❤♦♥♦r♠❛❧ ❜❛s❡ ✈❡❝t♦rs ❜② ei✱ i = 1, 2, 3✱ ❛♥② ♣♦✐♥t ✐♥ s♣❛❝❡ ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ✐♥ t❤❡ ❢♦r♠ r = xiei✱ ✇❤❡r❡ s✉♠♠❛t✐♦♥ ♦♥ r❡♣❡❛t❡❞ ✐♥❞✐❝❡s ✐s ✐♠♣❧✐❡❞✳ ❚❤❡ ❞❡❢♦r♠❡❞ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠ ✐s s♣❡❝✐✜❡❞ ❜② ❛ ❢✉♥❝t✐♦♥ r = r (R) = r (X, Y, S) ❣✐✈✐♥❣ t❤❡ ♣♦s✐t✐♦♥ ✐♥ s♣❛❝❡ ❝♦rr❡s♣♦♥❞✐♥❣ t♦ ❡❛❝❤ ♠❛t❡✲ r✐❛❧ ♣♦✐♥t R ❛t t❤❡ ❞❡❢♦r♠❡❞ ❝♦♥✜❣✉r❛t✐♦♥ s♦ t❤❛t xi = xi(Xp). ❋♦r s✐♠♣❧✐❝✲ ✐t②✱ ✐t ✐s ❛ss✉♠❡❞ t❤❛t ❆ss✉♠♣t✐♦♥ ✵✳ r(0) = 0✱ ❛♥❞ t❤❡ ♣♦✐♥ts (X, Y, 0) ❛r❡ ♠❛♣♣❡❞ ✐♥t♦ (a1X, a2Y, 0)✱ a1, a2 > 0✳ ■♥ ❛♥❛❧♦❣② ✇✐t❤ t❤❡ ♥♦t❛t✐♦♥ ✇❡ ✐♥tr♦❞✉❝❡❞ ❡❛r❧✐❡r✱ t❤❡ ❝✉r✈❡ r0(S) = r(0, 0, S) ✐♥ t❤❡ ♣❤②s✐❝❛❧ s♣❛❝❡ ✇✐❧❧ ❞❡♥♦t❡ t❤❡ ❝❡♥t❡r❧✐♥❡ ❝✉r✈❡ ❛t t❤❡ ❞❡✲ ❢♦r♠❡❞ st❛t❡✳ ❲❡ ♥♦✇ ♠❛❦❡ t❤❡ ❜❛s✐❝ ❛ss✉♠♣t✐♦♥s r❡❣❛r❞✐♥❣ t❤❡ ❦✐♥❡♠❛t✐❝s ♦❢ t❤❡ ❛r♠✳ ❚❤❡s❡ ❛ss✉♠♣t✐♦♥s s❧✐❣❤t❧② ❣❡♥❡r❛❧✐③❡ tr❛❞✐t✐♦♥❛❧ ❊✉❧❡r✲❇❡r♥♦✉❧❧✐ ♣♦st✉❧❛t❡s ❢♦r r♦❞ t❤❡♦r② ✐♥ s♦❧✐❞ ♠❡❝❤❛♥✐❝s ✇❤❡r❡ ♥♦✇ ✐♥✲♣❧❛♥❡ ❞❡❢♦r♠❛t✐♦♥s ♦❢ t❤❡ ❝r♦ss s❡❝t✐♦♥s ❛r❡ ❛❞♠✐ss✐❜❧❡✳ ❆ss✉♠♣t✐♦♥ ✶✳ ❋♦r ❡❛❝❤ S0 ∈ [0, 1]✱ t❤❡ ❡❧❧✐♣s❡ {(X, Y, S0), X2/a2
0+Y 2/b2 0 ≤
1} r❡♣r❡s❡♥t✐♥❣ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❛r♠ ❛t S0✱ ✐s ♠❛♣♣❡❞ ♦♥t♦ ❛♥ ❡❧❧✐♣s❡ ❝❡♥t❡r❡❞ ❛t r0(S0)✳ ❆ss✉♠♣t✐♦♥ ✷✳ ❚❤❡ ❡❧❧✐♣s❡ ❝♦♥t❛✐♥✐♥❣ t❤❡ ♣♦✐♥ts r(X, Y, S0) ✐s ♣❡r♣❡♥❞✐❝✲ ✉❧❛r t♦ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ❛t r0(S0)✱ ✐✳❡✳✱ (r(X, Y, S0) − r0(S0)) · dr0 dS (S0) = 0 ✭✸✳✶✮ ❢♦r ❛❧❧ X, Y ✳ ❆ss✉♠♣t✐♦♥ ✸✳ ❱❡❝t♦rs ✐♥ t❤❡ ♣❧❛♥❡ {(X, Y, S0)} ❛r❡ ♠❛♣♣❡❞ ❧✐♥❡❛r❧② t♦ t❤❡ ♣❧❛♥❡ ♦❢ t❤❡ ❡❧❧✐♣s❡ ❛t r0(S0)✱ ✐✳❡✳✱ ❢♦r ❡❛❝❤ S0 t❤❡ ♠❛♣♣✐♥❣ R(X, Y, S0) − R0(S0) → (r(X, Y, S0) − r0(S0)) ✭✸✳✷✮ ✐s ❧✐♥❡❛r✳ ❆ss✉♠♣t✐♦♥ ✹✳ ❚❤❡ ❧✐♥❡s {(X, 0, S0)} ❛♥❞ {(0, Y, S0)} ❛r❡ ♠❛♣♣❡❞ t♦ t❤❡ ♣r✐♥❝✐♣❛❧ ❛①❡s ♦❢ t❤❡ ❡❧❧✐♣s❡ r(X, Y, S0)✳ ❲❡ ✇✐❧❧ ♥❛t✉r❛❧❧② r❡❢❡r t♦ t❤❡ ♣♦✐♥ts r(X, Y, S0) ❛s t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠ ❛t S0. ✸✳✷✳ ❚❤❡ ❈❡♥t❡r❧✐♥❡ ❚r✐❛❞s✳ ❋♦r ❡❛❝❤ ♣♦✐♥t ✐♥ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠✱ ❝♦♥s✐❞❡r t❤❡ ❜❛s❡ ✈❡❝t♦rs gp = ∂r ∂Xp ✭✸✳✸✮ ❛♥❞ ♥♦t❡ t❤❛t gp = ∂r ∂Xp = ∂r ∂xi ∂xi ∂Xp = ∂xi ∂Xp ei. ✭✸✳✹✮
SLIDE 6
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✻
❚❤❡ ✈❡❝t♦r gp ❛t t❤❡ ♣♦✐♥t r1 = r(R1) ✐s t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡ t❤r♦✉❣❤ r1 t❤❛t ❝♦♥t❛✐♥s t❤❡ ✐♠❛❣❡ ♦❢ t❤❡ ❝✉r✈❡ R(Xp) = R1 + X(p)G(p) ✭♥♦ s✉♠♠❛t✐♦♥✮✳ ❚❤✉s ❢♦r ❡①❛♠♣❧❡✱ g3(0, 0, S) = ∂r ∂S (0, 0, S) = dr0 dS (S) ✭✸✳✺✮ ✐s t❤❡ t❛♥❣❡♥t ✭♥♦t ♥❡❝❡ss❛r✐❧② ♦❢ ✉♥✐t ❧❡♥❣t❤✮ t♦ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ r0(S)✳ ■♥ ❛❞❞✐t✐♦♥✱ t❤❡ ✈❡❝t♦rs g1 ❛♥❞ g2 ❛r❡ t❛♥❣❡♥t t♦ t❤❡ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠✳ ❋r♦♠ ❆ss✉♠♣t✐♦♥s ✸ ❛♥❞ ✹ ✐t ❢♦❧❧♦✇s t❤❛t t❤❡ ❜❛s❡ ✈❡❝t♦rs g1, g2 ❛r❡ ✉♥✐✲ ❢♦r♠ ❛♥❞ ♠✉t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r ✐♥ ❛♥② ♣❛rt✐❝✉❧❛r ❝r♦ss✲s❡❝t✐♦♥✳ ■♥ ❡❛❝❤ ❡❧❧✐♣t✐❝❛❧ ❝r♦ss s❡❝t✐♦♥ ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠✱ g1 ❛♥❞ g2 ❛r❡ ♣❛r❛❧❧❡❧ t♦ t❤❡ ♣r✐♥❝✐♣❛❧ ❛①❡s✳ ❚❤❡s❡ t✇♦ ✈❡❝t♦rs r❡♣r❡s❡♥t t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ t✇♦ ♠✉✲ t✉❛❧❧② ♣❡r♣❡♥❞✐❝✉❧❛r tr❛♥s✈❡rs❡ ♠✉s❝❧❡ ❣r♦✉♣s ✐♥ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠✳ ❇② ❆ss✉♠♣t✐♦♥ ✷✱ g3(0, 0, S) ✐s ♣❡r♣❡♥❞✐❝✉❧❛r t♦ ❜♦t❤ g1 ❛♥❞ g2✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ tr✐❛❞s gp(0, 0, S) ❝♦♥t❛✐♥ ♠✉t✉❛❧❧② ♦rt❤♦❣♦♥❛❧ ✈❡❝t♦rs✳ ❚❤❡ ✈❡❝t♦rs gp(X, Y, S) ❛t ♣♦✐♥ts ♦t❤❡r t❤❛♥ t❤❡ ❝❡♥t❡r❧✐♥❡ ♥❡❡❞ ♥♦t ❜❡ ♣❡r♣❡♥❞✐❝✉❧❛r✳ ■❢✱ ❢♦r ❡①❛♠♣❧❡✱ t❤❡ ❞❡❢♦r♠❡❞ ❛r♠ ❜❡❝♦♠❡s ❝♦♥✐❝❛❧✱ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧ ✜❜❡rs ❛r❡ ♥♦ ❧♦♥❣❡r ♣❛r❛❧❧❡❧✳ ■t ✐s ♥♦t❡❞ t❤❛t t❤❡ ❜❛s❡ ✈❡❝t♦rs ❛r❡ ♥♦t ♥❡❝❡ss❛r✐❧② ♦❢ ✉♥✐t ❧❡♥❣t❤ ❞✉❡ t♦ t❤❡ ❝❡♥t❡r❧✐♥❡ ❡①t❡♥s✐♦♥ ❛♥❞ t❤❡ ❝❤❛♥❣❡ ✐♥ t❤❡ ♣r✐♥❝✐♣❛❧ ❛①❡s ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝r♦ss✲s❡❝t✐♦♥✳ ❲❡ ✇✐❧❧ r❡❢❡r t♦ t❤❡ tr✐❛❞s gp(0, 0, S) ❛s t❤❡ ❝❡♥t❡r❧✐♥❡ tr✐❛❞s✳ ■t ❢♦❧❧♦✇s ❢r♦♠ ❊q✉❛t✐♦♥ ✭✸✳✹✮ t❤❛t ❛t ❡❛❝❤ S t❤❡r❡ ✐s ❛ ❧✐♥❡❛r ♠❛♣♣✐♥❣ T(S) ✇❤♦s❡ ♠❛tr✐① ✐s ∂xi/∂Xp(0, 0, S) s✉❝❤ t❤❛t gp(0, 0, S) = T(S)ipei. ✭✸✳✻✮ ■t ✐s r❡❝❛❧❧❡❞ t❤❛t ❛❝❝♦r❞✐♥❣ t♦ t❤❡ ♣♦❧❛r ❞❡❝♦♠♣♦s✐t✐♦♥ t❤❡♦r❡♠✱ ❛ ♥♦♥✲ s✐♥❣✉❧❛r ❧✐♥❡❛r ♠❛♣♣✐♥❣ T ♠❛② ❜❡ ❞❡❝♦♠♣♦s❡❞ ✐♥ t❤❡ ❢♦r♠ T = Q ◦ U ✭✸✳✼✮ ✇❤❡r❡ Q ✐s ❛♥ ♦rt❤♦❣♦♥❛❧ ♠❛♣♣✐♥❣ ❛♥❞ U ✐s ❛ ♣♦s✐t✐✈❡ ❞❡✜♥✐t❡ s②♠♠❡tr✐❝ ♠❛♣♣✐♥❣✳ ❆♣♣❧②✐♥❣ t❤✐s t♦ t❤❡ ♠❛♣♣✐♥❣s T(S)✱ s♦ T(S) = Q(S) ◦ U(S)✱ ♦♥❡ ❝❛♥ ✇r✐t❡ ❢♦r t❤❡ ❝❡♥t❡r❧✐♥❡ tr✐❛❞s gp(0, 0, S) = Q(S)ijU(S)jpei. ✭✸✳✽✮ ❊❛❝❤ ♦❢ t❤❡ tr✐❛❞s {dj(S)}✱ ❞❡✜♥❡❞ ❜② dj(S) = Q(S)ijei, ✭✸✳✾✮ ❝♦♥t❛✐♥s ♠✉t✉❛❧❧② ♦rt❤♦❣♦♥❛❧ ✉♥✐t ✈❡❝t♦rs✳ ❆s t❤❡ ♣❛r❛♠❡t❡r S ✈❛r✐❡s✱ t❤❡ ♦rt❤♦♥♦r♠❛❧ tr✐❛❞ r♦t❛t❡s ❛❝❝♦r❞✐♥❣ t♦ Q(S) ✭s❡❡ ❋✐❣✉r❡ ✸✳✷✮✳ ■♥ ♦✉r ❝❛s❡✱ ❛s t❤❡ ✈❡❝t♦rs gp(0, 0, S) ❛r❡ ♠✉t✉❛❧❧② ♦rt❤♦❣♦♥❛❧✱ t❤❡ ♣♦❧❛r ❞❡❝♦♠♣♦s✐t✐♦♥ ✐s ♣❛rt✐❝✉❧❛r❧② s✐♠♣❧❡✳ ❚❤❡ ✈❡❝t♦rs dj ❛r❡ s✐♠♣❧② t❤❡ ✉♥✐t ✈❡❝t♦rs ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ✈❡❝t♦rs gj✳ ❚❤❡ ♠❛tr✐① Qij ❝♦♥t❛✐♥s t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ dj ❛♥❞ t❤❡ ♠❛tr✐① Ujp ✐s ❞✐❛❣♦♥❛❧ ❛♥❞ ❝♦♥t❛✐♥s t❤❡ ♥♦r♠s gp ♦❢ t❤❡ ✈❡❝t♦rs ❜❡❧♦♥❣✐♥❣ t♦ t❤❡ ❝❡♥t❡r❧✐♥❡ tr✐❛❞ ♦♥ ✐ts ❞✐❛❣♦♥❛❧✳ ❚❤❡ ✈❛r✐♦✉s {di} tr✐❛❞s
SLIDE 7
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✼
d1(s) d3(s + s0) d2(s + s0) d1(s + s0) d2(s) g2(s + s0) g1(s + s0) g3(s + s0) d3(s)
❋✐❣✉r❡ ✸✳✷✿ ❚❤❡ tr✐❛❞s {gi} ❛♥❞ {di}, i = 1, 2, 3✳
❛ss♦❝✐❛t❡❞ ✇✐t❤ t❤❡ ♣♦✐♥ts S ∈ [0, l] ❛❧♦♥❣ t❤❡ ❝❡♥t❡r❧✐♥❡✱ ✇✐❧❧ ❜❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡ ♦rt❤♦♥♦r♠❛❧ r♦❞ ❢r❛♠❡s✳ ■t ❢♦❧❧♦✇s t❤❛t dp(S) = 1 gpgp(0, 0, S) ✭♥♦ s✉♠♠❛t✐♦♥✮, ✭✸✳✶✵✮ t❤❡ ✉♥✐t ✈❡❝t♦r d3 ✐s t❛♥❣❡♥t t♦ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ❝✉r✈❡✱ ❛♥❞ g1(S) = a1(S)d1(S), a1(S) = g1(0, 0, S), ✭✸✳✶✶✮ g2(S) = a2(S)d2(S), a2(S) = g2(0, 0, S). ✭✸✳✶✷✮ ❯s✐♥❣ s ❢♦r t❤❡ ❛r❝ ❧❡♥❣t❤ ♣❛r❛♠❡t❡r ❢♦r t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ❛♥❞ ❛ss✉♠✲ ✐♥❣ ♥❛t✉r❛❧❧② t❤❛t s(S) ✐s ❛ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣ ❢✉♥❝t✐♦♥✱ ✐t ❢♦❧❧♦✇s ❢r♦♠ ❊q✉❛t✐♦♥ ✭✸✳✺✮ t❤❛t t❤❡ str❡t❝❤ ♦r ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❝❡♥t❡r❧✐♥❡ ✐s ❣✐✈❡♥ ❜② ds dS (S) = g3(0, 0, S). ✭✸✳✶✸✮ ❲❡ ❞❡♥♦t❡ t❤❡ str❡t❝❤ ♦❢ t❤❡ ❛r♠✬s ❝❡♥t❡r❧✐♥❡ ❜② λ(S) = g3(0, 0, S) ❛♥❞ t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ✐s l = ´ λdS✳ ❯s✐♥❣ t❤❡ ❝❡♥t❡r❧✐♥❡ tr✐❛❞s✱ ♦✉r ❛ss✉♠♣t✐♦♥s ✐♠♣❧② t❤❛t t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠ ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ❜② r(R) = r0(S) + Xg1(0, 0, S) + Y g2(0, 0, S), = r0(S) + Xa1(S)d1(S) + Y a2(S)d2(S). ✭✸✳✶✹✮ ✸✳✸✳ ❚❤❡ ❊①t❡♥❞❡❞ ❉❛r❜♦✉① ❱❡❝t♦r✳ ❆s t❤❡ ♣❛r❛♠❡t❡r s ✈❛r✐❡s✱ t❤❡ tr✐❛❞ {di} ✉♥❞❡r❣♦❡s ❛ r✐❣✐❞ ♠♦t✐♦♥✳ ❚❤❡ ♦r✐❣✐♥ ♦❢ t❤❡ tr✐❛❞ ✐s ❞✐s♣❧❛❝❡❞ t❛♥❣❡♥t t♦ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡✳ ❚❤❡ ✈❡❝t♦rs di ❛r❡ r♦t❛t❡❞ r✐❣✐❞❧② ❛s ❡①♣r❡ss❡❞ ❜② ❊q✉❛t✐♦♥ ✭✸✳✾✮✳ ❆s s(S) ✇❛s ❛ss✉♠❡❞ t♦ ❜❡ ♠♦♥♦t♦♥✐❝❛❧❧② ✐♥❝r❡❛s✐♥❣✱ ♦♥❡ ♠❛② ❝♦♥s✐❞❡r t❤❡ ❞❡♣❡♥❞❡♥❝❡ di(s) = di(S(s))✳ ❈♦♥s✐❞❡r t❤❡ r❛t❡s ddi ds = ddi dS dS ds = 1 λ dQ(S)ji dS ej. ✭✸✳✶✺✮
SLIDE 8
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✽
❚❤❡s❡ r♦t❛t✐♦♥ r❛t❡s ♠❛② ❜❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ✈❡❝t♦r u s♦ t❤❛t✱ ddi ds = u × di. ✭✸✳✶✻✮ ❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ u ♠❛② ❜❡ ❢♦✉♥❞ ❜② ❞♦t ♠✉❧t✐♣❧②✐♥❣ ✭✸✳✶✻✮ ❜② dj ♦❜t❛✐♥✐♥❣ ddi ds · dj = (u × di) · dj. ✭✸✳✶✼✮ ❯s✐♥❣ ǫijk t♦ ❞❡♥♦t❡ t❤❡ ♣❡r♠✉t❛t✐♦♥ s②♠❜♦❧✱ ✇❡ ❤❛✈❡ ddi ds · dj = umǫijm, um = 1 2ǫijm ddi ds · dj. ✭✸✳✶✽✮ ■t ✐s str❛✐❣❤t❢♦r✇❛r❞ t♦ ✇r✐t❡ s✐♠✐❧❛r ❡①♣r❡ss✐♦♥s ❢♦r t❤❡ r❛t❡s r❡❧❛t✐✈❡ t♦ t❤❡ ♣❛r❛♠❡t❡r S ❛♥❞ ✇r✐t❡ t❤❡ r❡❧❛t✐♦♥s ❜❡t✇❡❡♥ t❤❡ t✇♦ t②♣❡s ♦❢ r❛t❡s✳ ■t ✐s ❝✉st♦♠❛r② ✐♥ r♦❞ t❤❡♦r② ✭s❡❡ ❬✶✸❪✮ t♦ ❞❡♥♦t❡ t❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r u ❛s {κ, κ′, τ}T s♦ κ κ′ τ = u1 u2 u3 = dd2 ds · d3 dd3 ds · d1 dd1 ds · d2 . ✭✸✳✶✾✮ ❉❡♥♦t✐♥❣ ❞✐✛❡r❡♥t✐❛t✐♦♥ ✇✐t❤ r❡s♣❡❝t t♦ S ❜② ❛ ♣r✐♠❡✱ ✇❡ ✐♠♠❡❞✐❛t❡❧② ❣❡t ❜② t❤❡ ❝❤❛✐♥ r✉❧❡ d′
i = λddi
ds = λu × di. ✭✸✳✷✵✮ ❚❤❡ ❧✐♥❡❛r ♠❛♣♣✐♥❣ Ω ❞❡✜♥❡❞ ❜② Ω(v) = λu × v ✭✸✳✷✶✮ ✐s r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♠❛tr✐① λ τ −κ′ −τ κ κ′ −κ . ✭✸✳✷✷✮ ❚❤✉s✱ ♦♥❡ ❤❛s d′
1
d′
2
d′
3
= Ω d1 d2 d3 . ✭✸✳✷✸✮ ❚❤❡ ❝♦♠♣♦♥❡♥ts ♦❢ t❤❡ ✈❡❝t♦r u ♠❛② ❜❡ ✐♥t❡r♣r❡t❡❞ ❛s ❢♦❧❧♦✇s✿ κ, κ′ r❡♣r❡s❡♥t t❤❡ ❜❡♥❞✐♥❣ ♦❢ t❤❡ ❝❡♥t❡r❧✐♥❡ ❛❜♦✉t t❤❡ ❛①❡s d1 ❛♥❞ d2✱ r❡s♣❡❝t✐✈❡❧②✱ ❛♥❞ τ ✐s t❤❡ t♦rs✐♦♥ ❛❜♦✉t t❤❡ t❛♥❣❡♥t t♦ t❤❡ ❝✉r✈❡✱ d3✳ ❚❤❡ ♣❛r❛♠❡t❡r τ ✐s ❞✐✛❡r❡♥t ❢r♦♠ t❤❡ ✐♥tr✐♥s✐❝ t♦rs✐♦♥ ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ✭❞❡s❝r✐❜❡❞ ✐♥ t❤❡ ♥❡①t s✉❜s❡❝t✐♦♥✮ ❛s ✐t ❛❝❝♦✉♥ts ❢♦r t❤❡ r❡❧❛t✐✈❡ t✇✐st ♦❢ t❤❡ ✈❛r✐♦✉s ❝r♦ss s❡❝t✐♦♥s ♦❢ t❤❡ ❛r♠✳ ■♥ ❛❞❞✐t✐♦♥✱ ✇❤✐❧❡ t❤❡ ✐♥tr✐♥s✐❝ t♦rs✐♦♥ ♦❢ ❛ ❝✉r✈❡ ✐s ♥♦t ❞❡✜♥❡❞ ❢♦r t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❝✉r✈❛t✉r❡ ✈❛♥✐s❤❡s ✭s❡❡ ♥❡①t s✉❜s❡❝t✐♦♥✮✱
SLIDE 9
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✾
τ ✐s ❛❧✇❛②s ✇❡❧❧ ❞❡✜♥❡❞✳ ■t ✐s ♥♦t❡❞ t❤❛t t❤❡ r♦t❛t✐♦♥ r❛t❡ ✈❡❝t♦r u ✐s ❛♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❉❛r❜♦✉① ✈❡❝t♦r ✉s❡❞ ✐♥ ❞✐✛❡r❡♥t✐❛❧ ❣❡♦♠❡tr②✳ ❋r♦♠ t❤❡ r❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ✐♥ ❊q✉❛t✐♦♥ ✭✸✳✶✹✮✱ ❛s t❤❡ ❝❡♥t❡r❧✐♥❡ tr✐❛❞s ♠❛② ❜❡ ♦❜t❛✐♥❡❞ ❢r♦♠ t❤❡✐r ❞❡r✐✈❛t✐✈❡s t❤r♦✉❣❤ ✐♥t❡❣r❛t✐♦♥ ❛♥❞ ✉s✐♥❣ t❤❡ ✐♥✐t✐❛❧ ❝♦♥❞✐t✐♦♥s ❣✐✈❡♥ ❜② ❆ss✉♠♣t✐♦♥ ✵✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❢✉♥❝t✐♦♥ {κ(S), κ′(S), τ(S), λ(S), a1(S), a2(S)}✱ ❞❡✜♥❡s ✉♥✐q✉❡❧② t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ ❛♥ ❡①t❡♥s✐❜❧❡ r♦❞ ✉♥❞❡r t❤❡ ❛ss✉♠♣t✐♦♥s ♠❡♥t✐♦♥❡❞ ❡❛r✲ ❧✐❡r✳ ❋♦r ❡①❛♠♣❧❡✱ r0(S) = ˆ S
σ=0
λ(σ)d′
3(σ)dσ.
✭✸✳✷✹✮ ✸✳✹✳ ❘❡♣r❡s❡♥t❛t✐♦♥ ♦❢ t❤❡ ❆r♠✬s ❈♦♥✜❣✉r❛t✐♦♥ ❯s✐♥❣ t❤❡ ❋r❡♥❡t✲ ❙❡rr❡t P❛r❛♠❡t❡rs✳ ❆♥ ❛❧t❡r♥❛t✐✈❡ ❛♣♣r♦❛❝❤ t♦ t❤❡ ❛❜♦✈❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❛r♠✬s ❝♦♥✜❣✉r❛t✐♦♥ ✐s ❜❛s❡❞ ♦♥ t❤❡ ✇❡❧❧ ❦♥♦✇♥ ❋r❡♥❡t✲❙❡rr❡t ♣❛r❛♠❡t❡rs ✭❋❙✮ ❬✶✹❪ ❢♦r ❛ s♣❛t✐❛❧ ❝✉r✈❡ r❡♣r❡s❡♥t❡❞ ❜② ❛ ✈❡❝t♦r ❢✉♥❝t✐♦♥✱ r0(s) ∈ R3✱ ✇❤❡r❡ s ✐s t❤❡ ❛r❝ ❧❡♥❣t❤ ❛❧♦♥❣ t❤❡ ❝✉r✈❡✳ ■t ✐s r❡❝❛❧❧❡❞ t❤❛t ❢♦r t❤❡ ❝❛s❡ ♦❢ ♥♦♥✲✈❛♥✐s❤✐♥❣ ❝✉r✈❛t✉r❡✱ ❛ ✉♥✐q✉❡ ❋r❡♥❡t✲❙❡rr❡t ❢r❛♠❡ ❝❛♥ ❜❡ ❛ss♦❝✐❛t❡❞ ✇✐t❤ ❡❛❝❤ ♣♦✐♥t ♦♥ t❤❡ ❝✉r✈❡✳ ❚❤❡ ❋r❡♥❡t✲❙❡rr❡t ♦rt❤♦♥♦r♠❛❧ ❜❛s✐s ❛t ❛ ♣♦✐♥t S ✐s ❣✐✈❡♥ ❜②✱ T = dr0 ds , N = 1 κFS dT ds , B = T × N, ✭✸✳✷✺✮ ✇❤❡r❡ T, N ❛♥❞ B ❛r❡ r❡❢❡rr❡❞ t♦ ❛s t❤❡ t❛♥❣❡♥t✱ ♥♦r♠❛❧ ❛♥❞ ❜✐✲♥♦r♠❛❧ ✈❡❝t♦rs✱ r❡s♣❡❝t✐✈❡❧②✳ ✭❲❡ ♦♠✐tt❡❞ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦♥ S ❢♦r ❜r❡✈✐t②✳✮ ❚❤❡ ♣❛r❛♠❡t❡rs κfs✱ t❤❡ ❝✉r✈❛t✉r❡✱ ❛♥❞ τfs✱ t❤❡ t♦rs✐♦♥✱ ❛r❡ ❞❡✜♥❡❞ ❜② κFS =
- dT
ds
- ,
τFS = dN ds · B. ✭✸✳✷✻✮ ■t ❝❛♥ ❜❡ s❤♦✇♥ t❤❛t t❤❡ ❝✉r✈❛t✉r❡ ❢✉♥❝t✐♦♥ ❛♥❞ t❤❡ t♦rs✐♦♥ ❢✉♥❝t✐♦♥ ✉♥✐q✉❡❧② ❞❡✜♥❡ ❛♥ ✐♥❡①t❡♥s✐❜❧❡ s♣❛t✐❛❧ ❝✉r✈❡ ✉♣ t♦ ❛ r✐❣✐❞ ❜♦❞② ❞✐s♣❧❛❝❡♠❡♥t ❬✶✹❪✳ ❚❤❡ ❋r❡♥❡t✲❙❡rr❡t tr✐❛❞s s❛t✐s❢② t❤❡ ❞✐✛❡r❡♥t✐❛❧ ❡q✉❛t✐♦♥s dT ds = κFSN, dN ds = −κFST + τFSB dB ds = −τFSN. , ✭✸✳✷✼✮ ❚❤❡ ❋r❡♥❡t✲❙❡rr❡t tr✐❛❞s ❞❡s❝r✐❜❡ t❤❡ ❣❡♦♠❡tr② ♦❢ t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡✳ ■♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠ ❝♦♠♣❧❡t❡❧②✱ ✇❡ ♥❡❡❞ t♦ ❛❝❝♦✉♥t ❢♦r t❤❡ str❡t❝❤ λ✱ t❤❡ ✐♥✲♣❧❛♥❡ ❞❡❢♦r♠❛t✐♦♥ ❛♥❞ t❤❡ t✇✐st ♦❢ t❤❡ ❛r♠ ❛❜♦✉t t❤❡ ❝❡♥t❡r❧✐♥❡✳ ■♥ ♦r❞❡r t♦ ❞❡s❝r✐❜❡ t❤❡ t✇✐st✱ ✇❡ ❞❡✜♥❡ ❛ ♣❛r❛♠❡t❡r φ✱ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ t❤❡ ♥♦r♠❛❧ ✉♥✐t ✈❡❝t♦r N ❛♥❞ t❤❡ ✐♠❛❣❡✱ g1✱ ♦❢ t❤❡ ✈❡❝t♦r G1 ✭s❡❡ ❋✐❣✉r❡ ✸✳✸✮✳
SLIDE 10
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✵
T N B N x φ X
Suctions
πs
(a) (b)
❋✐❣✉r❡ ✸✳✸✿ ✭❛✮ ❘❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ✭❜✮ ❈✉rr❡♥t ❝♦♥✜❣✉r❛t✐♦♥✳
- ✐✈❡♥ t❤❡ s❡t {κ, κ′, τ}✱ ♦♥❡ ❝❛♥ ✜♥❞ t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❋✳❙✳ ♣❛r❛♠❡t❡rs
❜②✱ κFS =
- κ′2 + κ2,
✭✸✳✷✽✮ τFS = τ + 1 (κ′2 + κ2)3/2
- κdκ′
ds − κ′ dκ ds
- ,
✭✸✳✷✾✮ φ = cos−1
- κ′
√ κ′2 + κ2
- = sin−1
- −κ
√ κ′2 + κ2
- .
✭✸✳✸✵✮ ✹✳ ❚❤❡ ❉❡❢♦r♠❛t✐♦♥ ●r❛❞✐❡♥t ❛♥❞ ❙tr❛✐♥ ✹✳✶✳ ❚❤❡ ▼❛tr✐① ♦❢ t❤❡ ❉❡❢♦r♠❛t✐♦♥ ●r❛❞✐❡♥t✳ ❊q✉❛t✐♦♥ ✭✸✳✶✹✮ ❢♦r t❤❡ ❞❡s❝r✐♣t✐♦♥ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ❞❡t❡r♠✐♥❡s t❤❡ ♣♦s✐t✐♦♥ ✈❡❝t♦r ✐♥ t❤❡ ❞❡❢♦r♠❡❞ st❛t❡ ♦❢ ❛ ♣❛rt✐❝❧❡ ❤❛✈✐♥❣ r❡❢❡r❡♥❝❡ ❝♦♦r❞✐♥❛t❡s (X, Y, S) ❜② r(X, Y, S) = xiei = r0(S) + Xg1(0, 0, S) + Y g2(0, 0, S). ❲❡ r❡❝❛❧❧ t❤❛t t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t ♦❢ s♦❧✐❞ ♠❡❝❤❛♥✐❝s✱ ✐s t❤❡ ❧✐♥❡❛r ♠❛♣♣✐♥❣ F = Fipei ⊗ Gp. ✭✹✳✶✮ r❡♣r❡s❡♥t❡❞ ❜② t❤❡ ♠❛tr✐① Fip = ∂xi ∂Xp . ✭✹✳✷✮ ❚❤✉s✱ t❤❡ ✜rst t✇♦ ❝♦❧✉♠♥s ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t ♠❛tr✐① ❛r❡ ❣✐✈❡♥ ❜② Fi1ei = ∂r ∂X = g1, ✭✹✳✸✮ Fi2ei = ∂r ∂Y = g2, ✭✹✳✹✮
SLIDE 11
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✶
❛♥❞ t❤❡ t❤✐r❞ ❝♦❧✉♠♥ ✐s ❣✐✈❡♥ ❜② Fi3ei = ∂r0 ∂S + X ∂g1 ∂S + Y ∂g2 ∂S = λd3 + X da1 dS d1 + a1 dd1 dS
- + Y
da2 dS d2 + a2 dd2 dS
- . ✭✹✳✺✮
❋♦r ❛♥② ♣❛rt✐❝✉❧❛r S✱ ♦♥❡ ♠❛② ❝❤♦♦s❡ t❤❡ ❜❛s✐s {ei} ✐♥ s♣❛❝❡ t♦ ❜❡ ✐❞❡♥t✐❝❛❧ t♦ t❤❡ tr✐❛❞ {di(S)}✳ ❯♥❞❡r t❤✐s s♣❡❝✐✜❝ ❝❤♦✐❝❡✱ t❤❡ ❧❛st ❡①♣r❡ss✐♦♥s ✐♠♣❧② t❤❛t t❤❡ ♠❛tr✐① ♦❢ F(X, Y, S) ❛ss✉♠❡s t❤❡ ❢♦r♠ [F]d (X, Y, S) = a1
da1 dS X − τ a2λY
a2 τ a1λX + da2
dS Y
λ− κ′a1λX + κa2λY , ✭✹✳✻✮ ✇❤❡r❡ t❤❡ ❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡ ✈❛r✐♦✉s ✈❛r✐❛❜❧❡s ♦♥ S ✇❛s ♦♠✐tt❡❞ ♦♥ t❤❡ r✐❣❤t✳ ✹✳✷✳ ❚❤❡ ❈♦♥s❡q✉❡♥❝❡s ♦❢ ■♥❝♦♠♣r❡ss✐❜✐❧✐t②✳ ❆s ♠❡♥t✐♦♥❡❞ ✐♥ ❙❡❝t✐♦♥ ✷✱ t❤❡ ♦❝t♦♣✉s✬s ❛r♠ ✐s ❛❧♠♦st ❡♥t✐r❡❧② ❝♦♠♣♦s❡❞ ♦❢ ✈✐rt✉❛❧❧② ✐♥❝♦♠♣r❡ss✐❜❧❡ ♠✉s❝❧❡ t✐ss✉❡✳ ■♥❞❡❡❞✱ ✐♥ ❡❛r❧✐❡r tr❡❛t♠❡♥ts ♦❢ ❖❝t♦♣✉s ❛r♠ ❦✐♥❡♠❛t✐❝s ✭❡✳❣✳✱ ❬✶✷❪✮ ✐t ✐s ❛ss✉♠❡❞ t❤❛t t❤❡ ❛r♠ ✐s ✐♥❝♦♠♣r❡ss✐❜❧❡✳ ❋♦r t❤❡ s❛❦❡ ♦❢ s✐♠♣❧✐❝✐t②✱ ✇❡ ❛ss✉♠❡ t❤❡ ✐♥❝♦♠♣r❡ss✐❜✐❧✐t② ❝♦♥str❛✐♥t ❤♦❧❞s ❢♦r s❡❣♠❡♥ts ♦❢ t❤❡ ❛r♠ r❛t❤❡r t❤❛♥ ♣♦✐♥t✇✐s❡✳ ❆ t❤❡♦r❡t✐❝❛❧ tr❡❛t♠❡♥t ♦❢ r♦❞ t❤❡♦r② ✇❤❡r❡ t❤❡ r♦❞ ✐s ❛ss✉♠❡❞ t♦ ❜❡ ♣♦✐♥t✇✐s❡ ✐♥❝♦♠♣r❡ss✐❜❧❡✱ ✇❛s ♣r❡s❡♥t❡❞ ♦♥❧② r❡❝❡♥t❧② ✐♥ ❬✶✵❪✳ ❚❤✉s✱ ✇❡ ♠❛❦❡ ❆ss✉♠♣t✐♦♥ ✺✳ ❚❤❡ ✈♦❧✉♠❡ ♦❢ ❛♥② s❡❣♠❡♥t✱ {(X, Y, S)}✱ 0 ≤ S1 ≤ S ≤ S2 ≤ 1✱ ♦❢ t❤❡ ❛r♠ ❞♦❡s ♥♦t ❝❤❛♥❣❡ ✉♥❞❡r ❞❡❢♦r♠❛t✐♦♥✳ ❈♦♥s✐❞❡r ❛ ✈♦❧✉♠❡ ❡❧❡♠❡♥t dV0 ❝♦♥t❛✐♥✐♥❣ ❛ ♠❛t❡r✐❛❧ ♣♦✐♥t R ❛♥❞ ✐ts ✐♠✲ ❛❣❡ dV ❝♦♥t❛✐♥✐♥❣ r(R)✳ ❚❤❡♥✱ ✉s✐♥❣ J✱ t❤❡ ❞❡t❡r♠✐♥❛♥t ♦❢ t❤❡ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t✱ ♦♥❡ ❤❛s dV/dV0 = J✳ ❚❤❡ ✈♦❧✉♠❡ V ♦❢ ❛ ❞❡❢♦r♠❡❞ s❡❣♠❡♥t ♦❢ t❤❡ ❛r♠ ✐s t❤✉s ❣✐✈❡♥ ❛s V = ˚ J dXdY dS , = ˆ S2
S1
a1(S)a2(S)λ(S)πa0b0 dS . ✭✹✳✼✮ ❆ss✉♠✐♥❣ t❤❛t t❤❡ ✐♥t❡❣r❛♥❞ ✐♥ ❊q✉❛t✐♦♥ ✭✹✳✼✮ ✐s ❝♦♥t✐♥✉♦✉s✱ ✇❡ ❝♦♥❝❧✉❞❡ t❤❛t ❛ ♥❡❝❡ss❛r② ❛♥❞ s✉✣❝✐❡♥t ❝♦♥❞✐t✐♦♥ ❢♦r t❤❡ ✈♦❧✉♠❡ ♦❢ ❡✈❡r② s❡❣♠❡♥t ♦❢ t❤❡ ❛r♠ t♦ r❡♠❛✐♥ ✉♥❝❤❛♥❣❡❞✱ ✐✳❡✳✱ t❤❛t V = V0 = πa0b0(S2 − S1)✱ ✐s✱ λ(S) = 1 a1(S) a2(S), ∀S ∈ [0, 1]. ✭✹✳✽✮ ❙✐♥❝❡ t❤❡ ❧❛st ❡q✉❛t✐♦♥ ❝❛♥♥♦t ❞❡t❡r♠✐♥❡ ❛ ✉♥✐q✉❡ ♣❛✐r (a1, a2) ✇❡ ♠❛❦❡ t❤❡ ❢♦❧❧♦✇✐♥❣ ❆ss✉♠♣t✐♦♥ ✻✳ ❚❤❡ ❛r♠ ♣r❡s❡r✈❡s t❤❡ ✐♥✐t✐❛❧ r❛t✐♦ ❜❡t✇❡❡♥ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ ♣r✐♥❝✐♣❛❧ ❛①❡s ♦❢ t❤❡ ❡❧❧✐♣t✐❝ ❝r♦ss✲s❡❝t✐♦♥✳
SLIDE 12
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✷
❲❡ ❞❡♥♦t❡ t❤❡ ❛❜♦✈❡ ♠❡♥t✐♦♥❡❞ r❛t✐♦ ❛s r := a0
b0 ✳ ❈♦♥s❡q✉❡♥t❧②✱ a0 b0 = a1a0 a2b0 ✱
❛♥❞ s♦✱ a1(S) = a2(S) = a(S). ✹✳✸✳ ❙tr❛✐♥ ❆♥❛❧②s✐s✳ ❈♦♥s✐❞❡r ❛♥ ✐♥✜♥✐t❡s✐♠❛❧ ✈❡❝t♦r dX = dXpGp ✭✹✳✾✮ ♦r✐❣✐♥❛t✐♥❣ ❛t t❤❡ ♣♦✐♥t R ✐♥ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ✇❤♦s❡ ✐♠❛❣❡ ✉♥❞❡r t❤❡ ❞❡❢♦r♠❛t✐♦♥ ✐s dx = dxiei = ∂xi ∂Xp dXpei = F(dX) ✭✹✳✶✵✮ ♦r✐❣✐♥❛t✐♥❣ ❛t r(R)✳ ■t ✐s ❝♦♥✈❡♥✐❡♥t✱ ❛♥❞ ✐♥❞❡❡❞ ♦❢ ✇✐❞❡ ✉s❡ ✐♥ t❤❡ ♠❡❝❤❛♥✐❝s ♦❢ ❝♦♥t✐♥✉♦✉s ♠❡❞✐❛✱ t♦ ❞❡s❝r✐❜❡ t❤❡ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❡❧❡♠❡♥t dX ❜② t❤❡ q✉❛♥t✐t② 1 2 [dx · dx − dX · dX] = 1 2
- FTF − I
- (dX) · dX = E (dX) · dX,
✭✹✳✶✶✮ ✇❤❡r❡✱ E = 1 2
- FTF − I
- ✭✹✳✶✷✮
✐s t❤❡ ▲❛❣r❛♥❣✐❛♥ str❛✐♥ t❡♥s♦r✳ ❋♦r t❤❡ ❝❛s❡ ✇❤❡r❡ t❤❡ ❞❡❢♦r♠❡❞ st❛t❡ ♦❢ t❤❡ ❛r♠ ❝❛♥ ❜❡ ♦❜t❛✐♥❡❞ ❜② s✉♣❡r✐♠♣♦s✐♥❣ ❛ s♠❛❧❧ ❞✐s♣❧❛❝❡♠❡♥t ✜❡❧❞ ♦♥ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥✱ ❛♥❞ t❤❡ ✈❡❝t♦r dX ✐s ♥♦r♠❛❧✐③❡❞ t♦ ❜❡ ♦❢ ✉♥✐t ❧❡♥❣t❤✱ 1
2 [dx · dx − dX · dX] ✐s t❤❡ ❧✐♥❡❛r ❛♣♣r♦①✐♠❛t✐♦♥ t♦ t❤❡ ❝❤❛♥❣❡ ✐♥
❧❡♥❣t❤ ♦❢ dX ❞✉r✐♥❣ t❤❡ ❞❡❢♦r♠❛t✐♦♥✳ ❚❤✉s✱ ❢♦r ❛ ✉♥✐t ✈❡❝t♦r ˆ n✱ ♦r✐❣✐♥❛t✐♥❣ ❛t (X, Y, S)✱ ✐t ✐s ♥❛t✉r❛❧ t♦ r❡❢❡r t♦ ǫˆ
n(X, Y, S) = (E(X, Y, S)ˆ
n) · ˆ n ✭✹✳✶✸✮ ❛s t❤❡ str❛✐♥ ❛t t❤❡ ♣♦✐♥t (X, Y, S) ✐♥ t❤❡ ❞✐r❡❝t✐♦♥ ♦❢ ˆ n✳ ❖♥❝❡ ❛❣❛✐♥✱ t❤❡ ▲❛❣r❛♥❣✐❛♥ str❛✐♥ t❡♥s♦r ❤❛s ❛ s✐♠♣❧❡r ❡①♣r❡ss✐♦♥ ✇❤❡♥ ✇r✐tt❡♥ r❡❧❛t✐✈❡ t♦ t❤❡ ♦rt❤♦♥♦r♠❛❧ r♦❞ ❢r❛♠❡✱ ❛♥❞ ✇❡ ❤❛✈❡ [E]d = 1 2 a2 − 1 a2 − 1 a da
dS X − aλY τ
- a
da
dS Y + aλXτ
- a
da
dS X − aλY τ
- a
da
dS Y + aλXτ
- (aY λκ − aXλκ′ + λ)2+
da
dS Y + aλXτ
2 + da
dS X − aλY τ
2− 1 . ✭✹✳✶✹✮ ✺✳ ▼❛♥✐♣✉❧❛t♦r ❑✐♥❡♠❛t✐❝ ❆♥❛❧②s✐s ■♥ t❤✐s s❡❝t✐♦♥ ✇❡ ❝♦♥s✐❞❡r t❤❡ ♦❝t♦♣✉s✬s ❛r♠ ❛s ❛ ♠❛♥✐♣✉❧❛t♦r ❛♥❞ ✇❡ st✉❞② ✐ts ❦✐♥❡♠❛t✐❝ ♣r♦♣❡rt✐❡s✱ s♣❡❝✐✜❝❛❧❧②✱ t❤❡ ✐♥✈❡rs❡ ❛♥❞ ❞✐r❡❝t ❦✐♥❡♠❛t✐❝s✳ ■♥ ♦r❞❡r t♦ ♣❡r❢♦r♠ s✉❝❤ ❛♥ ❛♥❛❧②s✐s✱ ♦♥❡ ❤❛s t♦ ❞❡✜♥❡ ✇❤❛t ♣❛r❛♠❡t❡rs ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥s s❤♦✉❧❞ ❜❡ ❝♦♥tr♦❧❧❡❞✳ ❚❤❡ ❛r♠ ✐s ✉s❡❞ ❛s ❛ t♦♦❧ ❛❧♦♥❣ ✐ts ❡♥t✐r❡ ❧❡♥❣t❤ ❛♥❞ t❤❡ ♦❜❥❡❝t✐✈❡ ✐s t♦ ❜r✐♥❣ t❤❡ s✉❝t✐♦♥ ❡❧❡♠❡♥ts ✐♥t♦ ❝♦♥t❛❝t ✇✐t❤ s♦♠❡ s✉r❢❛❝❡ ✐♥ s✉❝❤ ❛ ✇❛② t❤❛t t❤❡ ❛r♠ ❛♥❞ t❤❡ s✉r❢❛❝❡ ❛r❡ t❛♥❣❡♥t ❛❧♦♥❣
SLIDE 13
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✸
t❤❡ ❝♦♥t❛❝t ❧✐♥❡✳ ❚❤✉s✱ t❤❡ ♠❛♥✐♣✉❧❛t♦r ❦✐♥❡♠❛t✐❝ ❛♥❛❧②s✐s ✇✐❧❧ ❝♦♥s✐❞❡r t❤❡ ❝♦♥tr♦❧ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠ ❛s ❞❡s❝r✐❜❡❞ ❜② t❤❡ ❞❡❢♦r♠❡❞ ❝❡♥t❡r❧✐♥❡ ❛♥❞ ❣❡♥❡r❛❧✐③❡❞ ❉❛r❜♦✉① ✈❡❝t♦r ✭r❛t❤❡r t❤❛♥ ❥✉st t❤❡ ❡♥❞ ♦❢ t❤❡ ❛r♠ ♦r ❛ s❡❣♠❡♥t ♦❢ t❤❡ ❛r♠✱ ❢♦r ❡①❛♠♣❧❡✮✳ ❙♣❡❝✐✜❝❛❧❧②✱ s✉❝❤ ❛ ❝♦♥✜❣✉r❛t✐♦♥ ✇✐❧❧ ❜❡ ❣✐✈❡♥ ❜② t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s {κ(s), κ′(s), τ(s), λ(S)}✱ ✇❤❡r❡ ❊q✉❛t✐♦♥ ✭✹✳✽✮ ❛♥❞ ❆ss✉♠♣t✐♦♥ ✻ r❡❧❛t❡ t❤❡ ❡①t❡♥s✐♦♥ ♣❛r❛♠❡t❡r✱ λ(S)✱ ✇✐t❤ t❤❡ ❝r♦ss✲ s❡❝t✐♦♥ ♣❛r❛♠❡t❡r a(S)✳ ✺✳✶✳ ■♥✈❡rs❡ ❑✐♥❡♠❛t✐❝s✳ ❋♦r t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ♣r♦❜❧❡♠ t❤❡ ❝♦♥✜❣✲ ✉r❛t✐♦♥ ♦❢ t❤❡ ❖❝t♦♣✉s✬s ❛r♠ ✐s ❣✐✈❡♥ ✐♥ t❡r♠s ♦❢ t❤❡ ❢✉♥❝t✐♦♥s κ(s)✱ κ′(s)✱ τ(s), λ(S)✱ ❛♥❞ t❤❡ ❛❝t✉❛t✐♦♥ ✈❛r✐❛❜❧❡s ❛r❡ t❤❡ str❛✐♥s ✐♥ t❤❡ ✈❛r✐♦✉s ♠✉s❝❧❡ ❣r♦✉♣s✳ ■t ✇✐❧❧ ❜❡ ❛ss✉♠❡❞ t❤❛t t❤❡ ✜❜❡rs ♦❢ t❤❡ ✈❛r✐♦✉s ❣r♦✉♣s ❛r❡ ♣r❡s❡♥t ❝♦✐♥❝✐❞❡♥t❧② ❛t ❛❧❧ ♣♦✐♥ts ✐♥ t❤❡ ❛r♠✳ ❆❝❝♦r❞✐♥❣❧②✱ ✇❡ ✇✐❧❧ ❝❛❧❝✉❧❛t❡ t❤❡ str❛✐♥s ❛t ❡❛❝❤ ♣♦✐♥t ✐♥ t❤❡ ❛r♠ ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ✈❛r✐♦✉s ❣r♦✉♣s✳ ❲❡ s❡t ǫL✱ ǫT1✱ ǫT2✱ ǫH1✱ ǫH2 t♦ ❜❡ t❤❡ str❛✐♥s ✐♥ t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ ❧♦♥❣✐t✉❞✐♥❛❧✱ ♦r❛❧✲❛❜♦r❛❧ ❛♥❞ ❧❛t❡r❛❧ tr❛♥s✈❡rs❛❧✱ ❛♥❞ r✐❣❤t ❛♥❞ ❧❡❢t ❤❡❧✐❝♦✐❞❛❧ ❣r♦✉♣s✱ r❡s♣❡❝t✐✈❡❧②✳ ❚❤✉s✱ ǫL = d3 · E(d3), ǫT1 = d1 · E(d1), ǫT2 = d2 · E(d2), ǫH1 = ˆ nc · E(ˆ nc), ǫH2 = ˆ ncc · E(ˆ ncc), ✭✺✳✶✮ ✇❤❡r❡ ˆ nc ❛♥❞ ˆ ncc ❛r❡ ✉♥✐t ✈❡❝t♦rs ♣♦✐♥t✐♥❣ ❛t t❤❡ ❞✐r❡❝t✐♦♥s ♦❢ t❤❡ r✐❣❤t ❛♥❞ ❧❡❢t ❝♦✐❧❡❞ ❤❡❧✐❝♦✐❞❛❧ ♠✉s❝❧❡ ✜❜❡rs✱ r❡s♣❡❝t✐✈❡❧②✳ ■t ❛ss✉♠❡❞ t❤❛t ✐♥ t❤❡ r❡❢❡r❡♥❝❡ ❝♦♥✜❣✉r❛t✐♦♥ t❤❡ ❤❡❧✐❝♦✐❞❛❧ ✜❜❡rs ❛r❡ ❛t 45◦ ❛♥❣❧❡ t♦ t❤❡ ❝❡♥t❡r❧✐♥❡✷✳ ❚❤✉s✱ ˆ nc =
- −rY
A , r−1X A , 1 √ 2 T , ˆ ncc = rY A , −r−1X A , 1 √ 2 T , ✭✺✳✷✮ ✇❤❡r❡ A = √ 2 √ r2Y 2 + r−2X2✳ ■t ✐s ♥♦t❡❞ t❤❛t ❜② ❆ss✉♠♣t✐♦♥ ✻✱ ǫT1 = ǫT2✱ ❛♥❞ s♦ ✐t ✐s ♥❛t✉r❛❧ t♦ ❞❡✜♥❡ t❤❡ ✈❡❝t♦r ✜❡❧❞ ǫ(X, Y, S) = {ǫT1(X, Y, S), ǫL(S), ǫH1(X, Y, S), ǫH2(X, Y, S)}T ✭✺✳✸✮ t❤❛t ❝♦♥t❛✐♥s t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ ❛♥❛❧♦❣ ♦❢ t❤❡ ❛❝t✉❛t✐♦♥ ✈❛r✐❛❜❧❡s ❝♦♥tr♦❧❧✐♥❣ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠✳ ❋♦r t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ♣r♦❜❧❡♠ ✇❡ s❡❡❦ ❛ ♠❛♣♣✐♥❣ Ψ t❤❛t ❛❝ts ♦♥ t❤❡ s❡t ♦❢ ❢✉♥❝t✐♦♥s {κ(s), κ′(s), τ(s), λ(S)} ❛♥❞ ❣✐✈❡s ǫ✱ s♦✱ ǫ(X, Y, S) = Ψ(κ(s), κ′(s), τ(s), λ(S), X, Y, S). ✭✺✳✹✮
✷❚❤❡ ❣❡♥❡r❛❧✐③❛t✐♦♥ t♦ ❛♥② ♦t❤❡r ♣✐t❝❤ ❛♥❣❧❡ ✐s str❛✐❣❤t❢♦r✇❛r❞✳
SLIDE 14
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✹
❇② ✉s✐♥❣ ❊q✉❛t✐♦♥ ✭✹✳✶✸✮ ✇❡ ✜♥❞ t❤❛t✱ ǫT1 ǫL ǫH1 ǫH2 = [A] E11 E33 E13 E23 , ✭✺✳✺✮ ✇❤❡r❡ [A] = 1 1
1 2 1 2
− sin θ cos θ
1 2 1 2
sin θ − cos θ , ✭✺✳✻✮ sin θ =
a2
0Y
√
b4
0X2+a4 0Y 2 ✱ ❛♥❞ cos θ =
b2
0X
√
b4
0X2+a4 0Y 2 ✳
❲❡ ❞❡✜♥❡ ❛ ♥♦♥✲❧✐♥❡❛r ❢✉♥❝t✐♦♥ h t❤❛t t❛❦❡s t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♣❛r❛♠❡t❡rs ❛♥❞ ❣✐✈❡s t❤❡ ❢♦✉r str❛✐♥ ❝♦♠♣♦♥❡♥ts (E11, E33, E13, E23) = h(κ, κ′, τ, λ)✳ ❇② ❊q✉❛t✐♦♥ ✭✹✳✶✹✮ ✇❡ ❤❛✈❡ h(κ, κ′, τ, λ) = 1 2 a2 − 1 (aλκY − aλκ′X + λ)2 + (a′Y + aλτX)2 + (a′X − aλτY )2 − 1 aa′X − τY aa′Y + τX . ✭✺✳✼✮ ❍❡♥❝❡✱ t❤❡ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝s ♠❛♣♣✐♥❣ Ψ ✐s ❣✐✈❡♥ ❜② Ψ = A ◦ h. ✭✺✳✽✮ ✺✳✷✳ ❋♦r✇❛r❞ ❑✐♥❡♠❛t✐❝s✳ ❚❤❡ ❢♦r✇❛r❞ ❦✐♥❡♠❛t✐❝ ♣r♦❜❧❡♠ ✐s ❝♦♥❝❡r♥❡❞ ✇✐t❤ t❤❡ ✐♥✈❡rs❡ Φ = Ψ−1 ♦❢ t❤❡ ♠❛♣♣✐♥❣ Ψ ❞❡✜♥❡❞ ❛❜♦✈❡✳ ■♥ t❤❡ ❡①tr❡♠❡ ❝❛s❡ ♦♥❡ ♠✐❣❤t ❡①♣❡❝t t❤❛t t❤❡ ❞♦♠❛✐♥ ♦♥ ✇❤✐❝❤ Ψ ✐s t❤❡ ❝♦❧❧❡❝t✐♦♥ ♦❢ ❛❧❧ ❝♦♥t✐♥✉♦✉s str❛✐♥ ✜❡❧❞s {(E11(X, Y, S), E33(X, Y, S), E13(X, Y, S), E23(X, Y, S))}. ✭✺✳✾✮ ❍♦✇❡✈❡r✱ t❤✐s ❝❛♥♥♦t ❤♦❧❞ tr✉❡ ❜❡❝❛✉s❡ ♦❢ t❤❡ ❝♦♠♣❛t✐❜✐❧✐t② r❡str✐❝t✐♦♥ ∂Fij ∂Xp = ∂Fip ∂Xj = ∂2xi ∂Xj∂Xp ✭✺✳✶✵✮ ❢♦r t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ❞❡❢♦r♠❛t✐♦♥ ❣r❛❞✐❡♥t✳ ❋✉rt❤❡r♠♦r❡✱ ✐t ✐s ❝❧❡❛r t❤❛t ❛ ❣❡♥❡r✐❝ ❝♦♥✜❣✉r❛t✐♦♥ ✐♥❞✉❝❡❞ ❜② ❛ ❝♦♠♣❛t✐❜❧❡ str❛✐♥ ✜❡❧❞ ♥❡❡❞ ♥♦t s❛t✐s❢② ♥❡❝❡ss❛r✐❧② t❤❡ ❛ss✉♠♣t✐♦♥s ✇❡ ♠❛❞❡ ✐♥ ❙✉❜s❡❝t✐♦♥ ✸✳✶✳ ❋♦r ❡①❛♠♣❧❡✱ ✉s✐♥❣ ❊q✉❛t✐♦♥ ✭✹✳✶✹✮ ✇❡ ✜♥❞ t❤❛t E13X + E23Y = a da dS (X2 + Y 2). ✭✺✳✶✶✮ ❚❤✉s✱ ♦✉r ❛♥❛❧②s✐s ♦❢ t❤❡ ❢♦r✇❛r❞ ❦✐♥❡♠❛t✐❝s ♦❢ t❤❡ ❛r♠ ✇✐❧❧ ❧❡❛❞ ✉s t♦ t❤❡ ❝♦♥❝❧✉s✐♦♥ t❤❛t t❤❡ ✈❛❧✉❡s ♦❢ t❤❡ str❛✐♥ ❛t t✇♦ ♣♦✐♥ts (X1, Y1, S0)✱ (X2, Y2, S0) ✐♥ ❛ ❝r♦ss s❡❝t✐♦♥ S0✱ t❤❛t ❛r❡ ♥♦t ♦♥ t❤❡ ❝❡♥t❡r❧✐♥❡✱ ❞❡t❡r♠✐♥❡ t❤❡ ✈❛❧✉❡s ♦❢ a(S0)✱ da/dS(S0)✱ κ(s0)✱ κ′(s0)✱ τ(s0) ❛♥❞ λ(S0)✱ ✇✐t❤ s♦♠❡ ❛❞❞✐t✐♦♥❛❧ ❝♦♥s✐st❡♥❝② ❝♦♥❞✐t✐♦♥s✳
SLIDE 15
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✺
◆♦t✐❝✐♥❣ t❤❛t t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ A ✐s s✐♥❣✉❧❛r ❛♥❞ ✉s✐♥❣ dE11 dS = d dS 1 2(a2 − 1) = aa′ ✭✺✳✶✷✮ ❛♥❞ ǫT1 = E11✱ ✇❡ ❤❛✈❡ ǫT1 ǫL ǫH1
dǫT 1 dS
= 1 1
1 2 1 2
− sin θ cos θ
X X2+Y 2 Y X2+Y 2
E11 E33 E13 E23 , ✭✺✳✶✸✮ ✇❤❡r❡ ♥♦✇ t❤❡ tr❛♥s❢♦r♠❛t✐♦♥ ✐s ✐♥✈❡rt✐❜❧❡✳ ❚❤❡ ✐♥✈❡rs❡ ♦❢ ❊q✉❛t✐♦♥ ✭✺✳✶✸✮ ✇✐❧❧ ❣✐✈❡ t❤❡ ✈❡❝t♦r {E11, E33, E13, E23} ✐♥ t❡r♠s ♦❢ t❤❡ ♠♦❞✐✜❡❞ str❛✐♥ ❢✉♥❝t✐♦♥s ✈❡❝t♦r✱ {ǫT1, ǫL, ǫH1, dǫT 1
dS }✳ ■♥ ♦r❞❡r
t♦ r❡♣r❡s❡♥t t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♣❛r❛♠❡t❡rs {κ, κ′, τ, λ, a1, a2} ✐♥ t❡r♠s ♦❢ t❤❡ str❛✐♥ ❢✉♥❝t✐♦♥s✱ ✇❡ ✉s❡ ❊q✉❛t✐♦♥ ✭✹✳✶✹✮ t♦❣❡t❤❡r ✇✐t❤ ❊q✉❛t✐♦♥ ✭✺✳✶✸✮ t♦ ♦❜t❛✐♥ a1 = a2 = a = √2ǫT1 + 1, ✭✺✳✶✹✮ τ = (a2
0 − b2 0)XY
b2
0X2 + a2 0Y 2 ǫ′ T1 +
- b4
0X2 + a4 0Y 2
2(b2
0X2 + a2 0Y 2) (2ǫH1 − ǫT1 − ǫL) ,
✭✺✳✶✺✮ κaY − κ′aX =
- 2ǫL − (a′Y + aτλX)2 − (a′X − aτλY )2 + 1
λ − 1. ✭✺✳✶✻✮ ❙✐♥❝❡ ♥♦♥❡ ♦❢ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♣❛r❛♠❡t❡rs ❛r❡ ❢✉♥❝t✐♦♥s ♦❢ X ♦r Y ✱ ✇❡ ✜♥❞ t❤❛t t❤❡ ❡①♣r❡ss✐♦♥s ♦♥ t❤❡ r✐❣❤t ❤❛♥❞ s✐❞❡s ♦❢ ❊q✉❛t✐♦♥s ✭✺✳✶✹✮ ❛♥❞ ✭✺✳✶✺✮ ❞❡♣❡♥❞ ♦♥❧② ♦♥ S✳ ❚❤✉s✱ t❤❡ ✐♥❞❡♣❡♥❞❡♥❝❡ ♦❢ t❤❡s❡ ❡①♣r❡ss✐♦♥s ♦♥ X ❛♥❞ Y ✱ ♦r✐❣✐♥❛t✐♥❣ ❢r♦♠ t❤❡ ❦✐♥❡♠❛t✐❝❛❧ ❛ss✉♠♣t✐♦♥s ♠❛❞❡✱ ♠❛② ❜❡ ✉s❡❞ ❛s ❝♦♥❞✐t✐♦♥s ❢♦r t❤❡ ✐♥✲♣❧❛♥❡ str❛✐♥ ✜❡❧❞s t♦ ❜❡ ❝♦♠♣❛t✐❜❧❡ ✇✐t❤ s♦♠❡ ❝♦♥✜❣✉✲ r❛t✐♦♥✳ ❚♦ ✜♥❞ τ, κ, ❛♥❞ κ′✱ ✇❡ ❡✈❛❧✉❛t❡ ❊q✉❛t✐♦♥s ✭✺✳✶✺✮ ❛♥❞ ✭✺✳✶✻✮ ❛t t✇♦ ♣♦✐♥ts ✐♥ ❛ ❝r♦ss✲s❡❝t✐♦♥✳ ❋♦r s✐♠♣❧✐❝✐t②✱ ✇❡ ❝❤♦♦s❡ t♦ ❡✈❛❧✉❛t❡ t❤❡ str❛✐♥ ❢✉♥❝t✐♦♥s ✐♥ ❊q✉❛t✐♦♥ ✭✺✳✶✺✮ ❛t X = a0, Y = 0✱ ❛♥❞ t❤✉s ✇❡ ♦❜t❛✐♥✱ τ(S) = 1 a0
- 2ǫH1(a0, 0, S) − ǫT1(a0, 0, S) − ǫL(a0, 0, S)
- .
✭✺✳✶✼✮ ❙❡tt✐♥❣ X = 0, Y = b0✱ ❛♥❞ X = a0, Y = 0✱ ❛❧t❡r♥❛t✐✈❡❧② ✐♥ ❊q✉❛t✐♦♥ ✭✺✳✶✻✮✱ ✇❡ ♦❜t❛✐♥ κ(S) =
- 2ǫL(0, b0, S) −
- a′2(S) + τ 2(S)λ(S)
- b2
0 + 1 − λ(S)
a(S)λ(S)b0 , ✭✺✳✶✽✮ κ′(S) = λ(S) −
- 2ǫL(a0, 0, S) −
- τ 2(S)λ(S) + a′2(S)
- a2
0 + 1
a(S)λ(S)a0 . ✭✺✳✶✾✮
SLIDE 16
❖◆ ❚❍❊ ❑■◆❊▼❆❚■❈❙ ❖❋ ❚❍❊ ❖❈❚❖P❯❙✬❙ ❆❘▼ ✶✻
❋✐❣✉r❡ ✻✳✶✿ ■❧❧✉str❛t✐♦♥ ♦❢ t❤❡ ❛r♠✬s ❝♦♥✜❣✉r❛t✐♦♥ ❞❡♣✐❝t❡❞ ❜② ❊q✉❛t✐♦♥ ✻✳✶
❚♦ ❝♦♥❝❧✉❞❡✱ ✇❡ r❡❝❛❧❧ ❢r♦♠ ❊q✉❛t✐♦♥ ✭✹✳✽✮ t❤❛t λ(S) = 1 a2(S). ✭✺✳✷✵✮ ✻✳ ❊①❛♠♣❧❡ ❚♦ ❞❡♠♦♥str❛t❡ t❤❡ ✉s❡ ♦❢ t❤❡ ♠♦❞❡❧ ✐♥ ❝♦♠♣✉t✐♥❣ t❤❡ str❛✐♥s ✐♥ t❤❡ ❞✐✛❡r❡♥t ♠✉s❝❧❡ ✜❜❡rs✱ ✇❡ ❝♦♥s✐❞❡r t❤❡ ❢♦❧❧♦✇✐♥❣ ❝♦♥✜❣✉r❛t✐♦♥ ♦❢ t❤❡ ❛r♠✱ κ(S) = 5S, κ′(S) = 2.7, , τ(S) = 0.5S, λ(S) = 1, ✭✻✳✶✮ ✐❧❧✉str❛t❡❞ ✐♥ ❋✐❣✉r❡ ✻✳✶✳ ❇② s✉❜st✐t✉t✐♥❣ ❊q✉❛t✐♦♥ ✭✻✳✶✮ ✐♥t♦ ❊q✉❛t✐♦♥s ✭✺✳✺✮✱ ✭✺✳✻✮✱ ✇❡ ♦❜t❛✐♥ t❤❡ ❢♦❧❧♦✇✐♥❣ str❛✐♥ ✜❡❧❞✱ ǫT1 ǫL ǫH1 ǫH2 =
1 2
- (1 − 2.7X + 5SY )2 + 0.25S2(X2 + Y 2) − 1
- 1
2(ǫL + 0.25 (a2
0Y 2+b2 0X2)
√
Y 2a4
0+X2b4
S)
1 2(ǫL − 0.25 (a2Y 2+b2X2)
√
Y 2a4
0+X2b4
S) . ✭✻✳✷✮ ❈♦♠♣✉t✐♥❣ t❤❡ ❝♦♥✜❣✉r❛t✐♦♥ ♣❛r❛♠❡t❡rs ✉s✐♥❣ ❊q✉❛t✐♦♥s ✭✺✳✶✹✮✱ ✭✺✳✶✼✮✕ ✭✺✳✷✵✮ ❛♥❞ t❤❡ str❛✐♥s ✐♥ ✭✻✳✷✮✱ ✇✐❧❧ r❡s✉❧t ✐♥ t❤❡ s❛♠❡ ❝♦♥✜❣✉r❛t✐♦♥ ♣❛r❛♠✲ ❡t❡rs ❣✐✈❡♥ ✐♥ ❊q✉❛t✐♦♥ ✭✻✳✶✮✳ ▼♦r❡♦✈❡r✱ ✐t ✐s r❡❛❞✐❧② s❤♦✇♥ t❤❛t ❊q✉❛t✐♦♥ ✭✺✳✶✻✮ ❤♦❧❞s ❢♦r ❛❧❧ (X, Y ) ∈ {X = αa0, Y = βb0| 0 ≤ α ≤ 1, 0 ≤ β ≤ 1}✳ ❘❡❢❡r❡♥❝❡s
❬✶❪ ❆✳ ❈❛♥t✐♥♦ ❛♥❞ ●✳ ❚✉r❦✱ ✏❆r♠s✿ P❤②s✐❝❛❧ s✐♠✉❧❛t✐♦♥ ❛♥❞ ❝♦♥tr♦❧ ♦❢ ♠✉s❝✉❧❛r ❤②✲ ❞r♦st❛ts✑✱ ✐♥ ◆❡✉r❛❧ ■♥❢♦r♠❛t✐♦♥ Pr♦❝❡ss✐♥❣ ❙②st❡♠s✱ ✷✵✵✺✳ ❬✷❪ ❋✳ ❋❛❤✐♠✐✱ ❍✳ ❆s❤r❛✜✉♦♥✱ ❛♥❞ ❈✳ ◆❛t❛r❛❥✱ ✏❆♥ ✐♠♣r♦✈❡❞ ✐♥✈❡rs❡ ❦✐♥❡♠❛t✐❝ ❛♥❞ ✈❡✲ ❧♦❝✐t② s♦❧✉t✐♦♥ ❢♦r s♣❛t✐❛❧ ❤②♣❡r✲r❡❞✉♥❞❛♥t r♦❜♦ts✑✱ ■❊❊❊ ❚r❛♥s❛❝t✐♦♥s ♦♥ ❘♦❜♦t✐❝s ❛♥❞ ❆✉t♦♠❛t✐♦♥✱ ✈♦❧✳ ✶✽✱ ♥♦✳ ✶✱ ♣♣✳ ✶✵✸✕✶✵✼✱ ✷✵✵✷✳ ❬✸❪ ❇✳ ❆✳ ❏♦♥❡s ❛♥❞ ■✳ ❉✳ ❲❛❧❦❡r✱ ✏❆ ♥❡✇ ❛♣♣r♦❛❝❤ t♦ ❏❛❝♦❜✐❛♥ ❢♦r♠✉❧❛t✐♦♥ ❢♦r ❛ ❝❧❛ss ♦❢ ♠✉❧t✐✲s❡❝t✐♦♥ ❝♦♥t✐♥✉✉♠ r♦❜♦ts✑✱ ✐♥ Pr♦❝❡❡❞✐♥❣s ♦❢ t❤❡ ■❊❊❊ ■♥t❡r♥❛t✐♦♥❛❧ ❈♦♥❢❡r❡♥❝❡ ♦♥ ❘♦❜♦t✐❝s ❛♥❞ ❆✉t♦♠❛t✐♦♥✱ ✷✵✵✺✳
SLIDE 17